Algorithmic correspondence for relevance logics, bunched implication logics, and relation algebras: the algorithm PEARL and its implementation (Technical Report)

Abstract

The theory and methods of algorithmic correspondence theory for modal logics, developed over the past 20 years, have recently been extended to the language L R of relevance logics with respect to their standard Routley-Meyer relational semantics. As a result, the non-deterministic algorithmic procedure PEARL (acronym for 'Propositional variables Elimination Algorithm for Relevance Logic') has been developed for computing first-order equivalents of formulas of the language L R in terms of that semantics. PEARL is an adaptation of the previously developed algorithmic procedures SQEMA (for normal modal logics) and ALBA (for distributive and nondistributive modal logics). It succeeds on all inductive formulas in the language L R , in particular on all previously studied classes of Sahlqvist-van Benthem formulas for relevance logic.In the present work we re-interpret the algorithm PEARL from an algebraic perspective, with its rewrite rules seen as manipulating quasi-inequalities interpreted over Urquhart's relevant algebras. This enables us to complete the part of the Sahlqvist-van Benthem theorem still outstanding from the previous work, namely the fact that all inductive L R -formulas are canonical, i.e., are preserved under canonical extensions of relevant algebras. Via the discrete duality between perfect relevant algebras and Routley-Meyer frames, this establishes the fact that all inductive L Rformulas axiomatise logics which are complete with respect to first-order definable classes of Routley-Meyer frames. This generalizes the "canonicity via correspondence" result in [43] for (what we can now recognise as) a certain special subclass of Sahlqvist-van Benthem formulas in the "groupoid" sublanguage of L R where fusion is the only connective. By extending L R with a unary connective for converse and adding the necessary axioms, our results can also be applied to bunched implication algebras and relation algebras.We then present an optimised and deterministic version of PEARL, which we have recently implemented in Python and applied to verify the first-order equivalents of a number of important axioms for relevance logics known from the literature, as well as on several new types of formulas. In the paper we report on the implementation and on some testing results.